Just browsing some old stuff in my office for other thing I found the following:
$230.$ (April, 1915) Proposed by E. B. Escott, Ann Arbor, Michigan.
Find three numbers such that their sum, the sum of their squares, and the sum of their cubes , shall be a cube.
Note.--W. D. Cairns says this problem, which was proposed in
L'Intermediaire in $1900$,
remains unsolved to date, even though it was reprinted in February, $1913$.